I was able to book the lowest category room at the Umstead using the FHR online booking system, which I believe I did about two months before our stay. Did you compare against the Umstead's website to confirm that there is availability for the lowest category rooms?
If the Exec Suite is the lowest you can book I would probably find another hotel. Your chance of getting upgraded will be much lower and the cost is too rich for my blood. The room is nice but not that nice! I had actually emailed the hotel and requested the Governor's Suite when I booked. I checked back a week before our visit and they had noted my request and planned to put me in there. Sadly, two days before we arrived the suite was booked.